* Abreviations

- RHS: Right hand side

* Lecture 1

** Example 1

*** Equations

2x - y = 0
-x + 2y = 3

*** Parts

**** Coefficients

[ 2 -1 ]
[ -1 2 ]

**** Unknowns

[ x ]
[ y ]

**** Right hand sides

[ 0 ]
[ 3 ]

*** Matrix form

Written as matrix form:

[ 2 -1 ] [ x ] = [ 0 ]
[ -1 2 ] [ y ] = [ 3 ]

Usually will name the parts:

   A       X   =   b

*** Row picture

- draw a line in a coordinate system per row
- look at where the lines intersect
- read the coordinates at the intersection point -> that is the solution to the system of equations

*** Column picture

  [ 2]      [-1]   [ 0]
x [  ] +  y [  ] = [  ]
  [-1]      [ 2]   [ 3]

Seeing the columns as vectors.

How to combine the vectors to find the right amount (right hand side)?
-> Linear combination! Most important operation in the course!

Find numbers x and y (the linear combination) to get the right amount (right hand side).

- draw the vectors in a coordinate system
- see how many times one needs to step in the direction of the vectors to get to the right hand side coordinates
- the times are the result for x and y

** Example 2

- 3 equations
- 3 unknowns

+2x -1y     = 0
-1x +2y -1z = -1
    -3y +4z = 4

    [+2 -1  0]     [ 0]
A = [-1 +2 -1] b = [-1]
    [ 0 -3 +4]     [+4]

*** Row picture

- 3d coordinate system
- all points that solve an equation will result in a plane of points
- intersection of 2 planes: usually a line, unless planes are equal
- intersection of 3 planes: 2 was a line, a line intersects with plane usually in one point, unless the line is on the plane
- -> there is a point where 3 planes intersect, except for special cases
  - parallel planes
  - equal planes (no new information!)
- difficult to see in a graphic where the intersection point it.
- quit the row picture!

*** Column picture

  [ 2]     [-1]     [ 0]   [ 0]
x [-1] + y [-2] + z [-1] = [-1]
  [ 0]     [-3]     [ 4]   [ 4]

- What combination of those 3 vectors will get the right hand side?
- draw vectors in 3d coordinate system
- how many times of each vector we need to get to the RHS point?
  - the RHS is already at 1z!
  - no need to take the other vectors!
  - -> 0x + 0y + 1z = b

Can we solve Ax = b for every b?

- Idea: As long as we have some amount of each of the components x y z in the coordinate system, I think there should be a solution for every b.
  - Only partially correct.
  - If the 3 columns of the matrix lie in the same plane, they cannot solve for every possible b!
  - -> We might need to check, whether the columns are in the same plane.
  - Vectors are in the same plane, if ???
    - The vectors need to be independent from each other.
    - What does that mean?
- What does Ax mean?
  - Trying to find a linear combination, a vector, which will result in the RHS.

*** Matix form

Ax = b

with:

    [+2 -1  0]     [ 0]
A = [-1 +2 -1] b = [-1]
    [ 0 -3 +4]     [+4]

How to multiply a matrix by a vector?

A·x = ?

Column wise:

/"one time the first column, two times the second column"/

[ 2  5]   [ 1]     [ 2]     [ 5]   [12]
[     ] · [  ] = 1 [  ] + 2 [  ] = [  ]
[ 1  3]   [ 2]     [ 1]     [ 3]   [ 7]

Or row wise (dot product view):

[ 2  5]   [ 1]   [1*2 + 2*5]   [12]
[     ] · [  ] = [         ] = [  ]
[ 1  3]   [ 2]   [1*1 + 2*3]   [ 7]

* Lecture 2

TODO
